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We prove existence results for the obstacle problem related to the porous medium equation. For sufficiently regular obstacles, we find continuous solutions whose time derivative belongs to the dual of a parabolic Sobolev space. We also employ the notion of weak solutions and show that for more general obstacles, such a weak solution exists. The latter result is a consequence of a stability property of weak solutions with respect to the obstacle. Mathematics Subject Classification

We show that every weak supersolution of a variable exponent p-Laplace equation is lower semicontinuous and that the singular set of such a function is of zero capacity if the exponent is logarithmically Hölder continuous. As a technical tool we derive Harnacktype estimates for possibly unbounded supersolutions.

Abstract:This article describes modal analysis of acoustic waves in the human vocal tract while the subject is pronouncing ͓ø b ͔. The model used is the wave equation in three dimensions, together with physically relevant boundary conditions. The geometry is reconstructed from anatomical MRI data obtained by other researchers. The computations are carried out using the finite element method. The model is validated by comparing the computed modes with measured data.

We prove the uniqueness of viscosity solutions to a differential equation involving the infinity-Laplacian with a variable exponent. We also derive a version of Harnack's inequality for this minimax problem.